Bivariate model

As the procedure consists in the evaluation of the quality of all the models with one variable (i.e. p univariate models), of all the models with two variables [i.e. p X (p - 1) bivariate models], up to all the possible models with k variables, the greatest disadvantage of this method is the extraordinary increase in the required computer time when p and k are quite large. In fact, the total number t of models is given by the relationship ... 

In order to improve the performance of the calibration model other information from the spectral data could be included. The absorbance at 2i, for example, is negatively correlated with tryptophan concentration and may serve to compensate for the interfering species present. Including A21 gives the bivariate model defined by... 

Although the bivariate model performs considerably better than the univariate model, as evidenced by the smaller residuals, the calibration might be improved further by including more spectral data. The question arises as to which data to include. In the limit of course, all data will be used and the model takes the form... 

Figure 12 True and predicted concentrations using the bivariate model with A, and A2,...

In summary, the bivariate model is based on the two variables of interest the size of the crack (maximum size of one structural component) and the rate of corrosion. As a result, the propagation of a crack is controlled by the combination of two state-dependent gamma processes ... 

The CFD model described above is adequate for particle clusters with a constant fractal dimension. In most systems with fluid flow, clusters exposed to shear will restructure without changing their mass (or volume). Typically restructuring will reduce the surface area of the cluster and the fractal dimension will grow toward d — 3, corresponding to a sphere. To describe restructuring, the NDF must be extended to (at least) two internal coordinates (Selomulya et al., 2003 Zucca et al., 2006). For example, the joint surface, volume NDF can be denoted by n(s, u x, t) and obeys a bivariate PBE. 

In the second step we must add the micromixing terms from the DQMOM model to Eqs. (133)—(135). Fiowever, as we discussed earlier, we need to keep in mind that micromixing conserves the moments of the NDF, and not the weights and abscissas (see Eq. 113). The micromixing model in environment n for the bivariate moments has the form... 

Following Tjoa and Biegler (1991) we have modeled P <5) as a bivariate likelihood distribution, a contaminated normal distribution as shown in Eq. (11.10) with... 

Wright, D. L., R. McGraw, and D. E. Rosner (2001). Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering particle populations. Journal of Colloid and Interface Science 236, 242-251. 

Fig. 28. Bivariate example of bayesian confidence ellipses and SIMCA one-component models in a case of complex distributions. The direction of SIMCA components is not the same as the main axes of the ellipses because of the separate scaling used in SIMCA...

In SIMCA the distribution of the object in the inner model space is not considered, so the probability density in the inner space is constant and the overall PD appears as shown in Figs. 29, 30 for the enlarged and reduced SIMCA models. In CLASSY, Kernel estimation is used to compute the PD in the inner model space, whereas the errors in the outer space are considered, as in SIMCA, uncorrelated and with normal multivariate distribution, so that the overall distribution, in the inner and outer space of a one-dimensional model, looks like that reported in Fig. 31. Figures 32, 33 show the PD of the bivariate normal distribution and Kernel distribution (ALLOC) for the same data matrix as used for Fig. 31. Although in the data set of French wines no really important differences have been detected between SIMCA (enlarged model), ALLOC and CLASSY, it seems that CLASSY should be chosen when the number of objects is large and the distribution on the components of the inner model space is very different from a rectangular distribution. 

This is, however, not the whole of the matter. The superstructure ordering of point defects the collection of interstitial ions along certain fines or sheets, as in Magneli s model for the precursor of his shear structures the temperature-dependent adjustment of composition of a nonstoichiometric phase at the boundary of the bivariant range the nucleation of a new phase of different stoichiometry—these depend on accumulating vacancies or interstitials in some regions of the crystal lattice at the expense of others. 

Cecchi, T., Pncciarelh, F., and Passamonti, P. Extended thermodynamic approach to ion interaction chromatography a mono- and bivariate strategy to model the inflnence of ionic strength. J. Sep. Sci. 2004, 27,1323-1332. 

When only one sohd phase is present, the total concentration of the solution is indeterminate, and may be altered by addition of the other simple salt or of the double salt. The solubility of a salt is therefore not affected by the addition of a salt with a common ion. For the graphical representation of the equihbria in a three-component system, it is convenient to use a three-dimensional system of coordinates, of which the axes are the temperature and the concentrations of the two simple salts. Each point in the space corresponds to a definite vapour pressure. Monovariant equilibria are represented by lines, and bivariant equihbria by surfaces in the space model. (See van t Hoff, Bildung und Spaltung von Doppelsalzen, Leipzig 1897 also van t Hoff u. Meyerhoffer, Zeitschr. /. physikcd. Chemie, 30, 64 (1899), and others. Experimental methods of determining the transition point are also described there.)... 

---